One of the best-received variants of four-handed chess is
Chessapeak Challenge (link to the commercial producer is now down)
which solves one of the major problems of the game by having pawns
change directions during their march.

Since I do not have a complete copy of the rules, I called this
applet, respectfully, "Chesapeare Four-Handed Chess". Here, I permitted
players to capture opponent's units. The game is over when a player
on turn cannot make a legal move. This is either checkmate (win for
the other side) or stalemate.